Definition

Write Ch•+,−,b(𝒜)↪Ch•(𝒜)Ch_\bullet^{+,-,b}(\mathcal{A}) \hookrightarrow Ch_\bullet(\mathcal{A}) for the full subcategory on the chain complexes which are, respectively, bounded above, bounded below or bounded.

Definition

Write K(𝒜)K(\mathcal{A}) for the category obtained from Ch•(𝒜)Ch_\bullet(\mathcal{A}) by identifying homotopic chain maps.